Applications to 3-D imaging

Since the layer-stripping algorithm is a Kirchhoff method, it is readily applicable to 3-D data sets. The most efficient mode of application will depend on the particular acquisition parameters and there will be intermediate data volumes. Many of the same sampling and coverage issues that arise in 3-D migration will be issues in 3-D datuming also. Depending on the acquisition geometry, the intermediate volume could turn out to be larger than the original volume because out-of-plane propagation must be accommodated and saved at intermediate depth levels.

In some cases, a common azimuth approach can be applied. This is easy to do with the Kirchhoff formulation, because it only amounts to 3-D downward continuation in what could be termed a target-oriented manner. If the cross-line lateral velocity variation is not too severe, marine data can be downward continued to a grid with the same geometry as the original acquisition geometry. In this case, some energy will be lost, but this may be acceptable.

As in the case of rugged topography, the handling of land data acquired by shooting into patches of receivers is fairly straightforward: source and receiver gathers can be extrapolated using reciprocity. In this case, source spacing will be a critical parameter in determining if the common receiver gathers will be adequately sampled.

There are several factors that can make this approach very attractive in three dimensions. All of these computational factors hold for 2-D data also, but their effects should be more dramatic in 3-D applications. The following items could offer significant efficiency in three dimensions:

• Traveltimes for 3-D imaging are usually calculated in spherical coordinates. For a given cube of desired traveltimes, this calculation requires a spherical volume with a radius equal to the longest cube diagonal. Therefore, if the velocity model is subdivided, the number of traveltimes that need to be calculated is dramatically reduced. The reduced volume also significantly reduces the storage space required during the calculation.
• After each depth step, the resynthesized data traces get progressively shorter. This means that the computation and the amount of storage per trace is reduced after each depth step.
• After each depth step, both the migration and datuming aperture can be reduced. This not only cuts down on the number of calculations, but it can be used to limit the amount of input data kept in memory for any given output location.
• As mentioned in Chapter , a key algorithmic difference between migration and wave-equation datuming is that each input trace is statically shifted rather than dynamically. This means that the computation can be done quickly and efficiently as a complex multiplication in the Fourier-domain, where the derivative operator is applied. This is the primary reason why an optimized Kirchhoff datuming algorithm is potentially much faster than Kirchhoff migration.

The layer-stripping migration algorithm has great potential for efficiency, and for some geometries it is more efficient than standard Kirchhoff migration.